document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209618>Question 1209618</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose f(x) = \frac{9}{5} x - 4 + x^2. Does f have an inverse? If so, find f^{-1}(0). If not, enter \"undef\".  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849736 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849736\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine if f(x) has an inverse and find f⁻¹(0) if it does:\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Check if f(x) is one-to-one:**\r<BR>\n" );
document.write( "\n" );
document.write( "A function has an inverse if and only if it is one-to-one (injective), meaning that every element in the range corresponds to exactly one element in the domain.  A simple way to check if a function is one-to-one is to see if it passes the horizontal line test.  If any horizontal line intersects the graph of the function more than once, the function is not one-to-one and does not have an inverse.\r<BR>\n" );
document.write( "\n" );
document.write( "Our function is f(x) = x² + (9/5)x - 4.  This is a quadratic function, and its graph is a parabola. Parabolas fail the horizontal line test (except for very special cases where the domain is restricted to only one side of the vertex).  Since this is a parabola, it is not one-to-one.\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Conclusion:**\r<BR>\n" );
document.write( "\n" );
document.write( "Since f(x) is not one-to-one, it does not have an inverse.\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, f⁻¹(0) is *undef*.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );